Method and system for generating conditions representative of real and complex zeros

ABSTRACT

Roots of a function f(z) of a single complex variable quantity z are determined by generating a first sequence of physical conditions, each member of the sequence representing approximations of a root quantity according to the recursive relation WHERE N IS AN INDEX OF THE CONDITIONS IN EACH SEQUENCE. The first sequence is terminated at a first root quantity. Subsequent sequences of the conditions are generated with elimination in each of the subsequent sequence of the effect of all earlier generated root quantities.

United States Patent [72] Inventors Carl A. Barlow,.lr.

Dallas; Eric L. Jones, Dallas, Tex. [21] Appl. No. 444,554 [22] FiledApr. 1,1965 [45] Patented Feb. 9, I971 [73] Assignee Texas InstrumentsIncorporated Dallas, Tex. a corporation of Delaware [54] METHOD ANDSYSTEM FOR GENERATING CONDITIONS REPRESENTATIVE OF REAL AND COMPLEXZEROS 21 Claims, 6 Drawing Figs.

[52] US. Cl 235/156, 235/150.4, 235/152, 235/184, 235/197 [51] Int. ClG06! 7/38, 006g 7/32 [50] Field ofSearch 235/152,156,180,184,181,182,197,l50,150.3,150.31

[56] References Cited UNITED STATES PATENTS 2,915,246 12/1959 Piety235/180 3,233,086 2/1966 Schwetman 235/180 OTHER REFERENCES IterativeComputation of Complex Roots,"

Primary Examiner- Eugene G. Botz Assistant Examiner-Felix D. GruberAttorneys-Samuel M. Mims, Jr., James 0. Dixon, Andrew M. Hassell, JohnE. Vandigriff, Harold Levine and Richards, Harris & Hubbard ABSTRACT:Roots of a function f(z) of a single complex variable quantity z aredetermined by generating a first sequence of physical conditions, eachmember of the sequence representing approximations of a root quantityaccording to the recursive relation where n is an index of theconditions in each sequence. The first sequence is terminated at afirstroot quantity. Subsequent sequences of the conditions are generated withelimination in each of the subsequent sequence of the effect of allearlier generated root quantities.

by P. A. Samuelson, Journal of Mathematics and Physics, 1949, Vol.

[,208 QMULT MULT SHEET 2 BF 3 l 206 MULT MULT PATENTED FEB 9mm METHODAND SYSTEM FOR GENERATING CONDITIONS REPRESENTATIVE OF REAL AND COMPLEXZEROS This invention relates to treatment of signals which havecharacteristics that may be representative of general mathe- 5 maticalfunctions. and, more particularly. to production of conditionsrepresentative of zeros, poles, and eigenvalues. In a more specificaspect. the invention relates to generation of conditions representativeof the real zeros of functions of a single variable. which functionsalso have one or more complex zeros.

The present invention provides a simple and yet powerful method forgenerating physical conditions representative of the solutions to two ofthe more common numerical problems. The method makes possible theefficient generation of conditions which represent the zeros of acomplex function, either transcendental or algebraic. of a complexvariable. In addition, it is applicable to the generation of conditionsrepresentative of eigenvalues of a general matrix in which the parametermay appear in any elements of the matrix in a basically unrestrictedway.

I-or convenience in the following description, the term roots" will beused to refer to the zeros of a function, recognizing that functionshave zeros while equations have roots. The method provides for allowingconditions representative of 25 real roots of a function which alsopossess complex roots to be found without the difficulties accompanyingthe well-known secant method.

A problem which occurs frequently is that of determining the roots of ageneral arbitrary function of a single complex variable flz), or of theequation A similar problem is involved in connection with thedetermination of poles of an arbitrary function. This may be thought ofas finding the zeros of its inverse.

A second species of problem which frequently occurs in all branches ofengineering involves the determination of those values of a complexparameter A for which nonnull solutions exist to a set of linearalgebraic equations:

or more concisely wherein:

This problem is generally referred to as the general eigenvalue problem,where the set {Ml H (M)Xs= contains the eigenvalues or characteristicnumbers and the are the associated eigenvectors or characteristicshapes.

In general, the elements H of the given matrix may each depend on A in arather arbitrary way; however, the most frequently occurring situationis that where HEA-fl where is the identity matrix, whose only nonzeroelements are ones along the main diagonal, and

is independent of A and usually possesses special symmetry properties.Most of the prior art numerical procedures thus far discovered are onlyappropriate to this special case. lt is the purpose of the presentinvention to provide an improved and powerful method for solution of thegeneral problem stated above, by systems not heretofore known.

The eigenvalue and root problems are highly related. as may be seen bycomparing the form of equations A and C. The method provided here isapplicable to the generation of physical conditions representative ofsolutions to both problems.

In accordance with the invention, physical conditions are generated fora function flz) of the single complex variable quantity z. A firstsequence of physical conditions is generated, each representingapproximations of a root quantity according to the recursive relationwhere n is an index of the conditions in each sequence. The firstsequence is terminated at a first root quantity. Subsequent sequences ofthe conditions are subsequently generated with elimination in each ofthe subsequent sequence of the effect of all earlier generated rootquantities.

In accordance with a further aspect of the invention, there is provideda root condition generating system for generating physical conditionsrelated to or definitive of roots of a complex function of a singleindependent variable. The system is adapted to cooperate with anevaluating unit which produces, on test output channels thereof, outputconditions representing two values of the complex function for twostarting conditions applied to test input channels. The root conditiongenerating system includes a loop with two input channels and two outputchannels adapted to be closed by connections to the output channels andinput channels, respectively, of the evaluating unit. Means are providedfor producing in the system output channels a pair of different sets ofconditions which are representative of two complex numbers forapplication to the evaluating unit. A system control unit actuates thesystem for repeated cycles in which conditions each representative ofcomplex numbers are generated, progressively approaching valuesrepresentative of a first root condition.

Comparison means are provided for comparing each system output conditionwith the next preceding system output condi-' tion and for haltingoperation when the change between successive system output conditions iswithin set limits.

Means are then provided for modifying, on successive cycles ofgeneration of other root conditions, the character of the outputconditions produced by the evaluating unit, the modification primarilybeing dependent upon conditions representing all earlier generated rootconditions to eliminate the effect of the existence of prior roots onsubsequent root condition generation.

Preferably. the loop has structure according to the relation and themodifying means has structure according to the relation wherein i= 1, 2,3,4, n, and is equal to the number of root conditions previouslydetermined and i symbolically represents the operation [(z-r )(zr)...(z-r,-)].

As will hereinafter be shown, operations for generating conditionsrepresentative of the roots of functions, of the poles of functions, andof the eigenvalue problem are of generally the same character, and thusthe term root condition will be used herein as to apply to all of theforegoing categories of operations.

For a more complete understanding of the present invention and forfurther objects and advantages thereof, reference may now be had to thefollowing description taken in conjunction with the accompanyingdrawings in which:

FIG. 1 illustrates a portion of a system for generating conditionsrepresentative of roots;

FIG. 2 illustrates that portion of the system for eliminating the effectof roots previously generated in the generation of subsequent roots;

FIG. 3 illustrates that portion of the system for selecting preferredstarting voltages;

FIG. 4 is a diagram illustrative of the operation of FIG. 3;

FIG. 5 illustrates a portion of the system of FIG. 1 modified forgeneration of voltages representative of poles; and

FIG. 6 diagrammatically illustrates the relationship between thedrawings of FIGS. 1, 2 and 3.

GENERAL POLYNOMIAL FUNCTION The present invention is directed to dataprocessing in which representations of physical phenomena may beproduced in any one of several different forms. For the purpose ofillustrating the invention, FIGS. 1 and 2 show a system for treatment ofsignals in electrical form for the generation of output signalsrepresentative of the real and complex roots of a signal which behavesin accordance with the polynomial:

The system of FIGS. 1 and 2 is representative of those systems which arecapable of performing the present method. While the present invention isof general application, the particular system for generating electricalconditions representative of the roots of equation 1 will be described.Then there .will be described the generation of conditionsrepresentative characterized by the relationship:

f( )nl 'f( )n2 That is, if two values for z are chosen as startingvalues then 2 may be produced by generating signals representing thesolution to equation 2 with signals represented byequations 3 and 4substituted therein to establish the following relationship: I

and

representing the solution to Equation 5.

The operations represented by equation 2 are of general application tocomplex functions to obtain roots. By suitable manipulation, poles ofcomplex functions, as well as eigenvalues of complicated matrices, maybe obtained.

In the study of oscillations of such signals as in the example ofequation I, it frequently becomes necessary to find the value of thefunction for a number of complex arguments. The operation of the systemof FIGS. 1 and 2 provides for the evaluation of the polynomial ofequation 1 for a series of complex arguments which are varied, byiteration, through a series of steps Z Z Z 2,, to produce a firstcomplex root z,,= r,. Thereafter, a second root r is obtained byoperation of the system of FIGS. 1 and 2 with signals modified inaccordance with:

l r- 1) and IIELL The third root signal r;, is then generated with thesignals modified in accordance with:

Signals representative of each of the three roots r,, r and r thus arereadily obtainable.

FIGS. 1 AND 2 The circuit of FIGS. 1 and 2 will now be described alongwith its operation to produce signals representative of the three rootsr r and r Referring now to FIGS. 1 and 2, the system will store systemoutput signals on or in a suitable storage medium such as on a.

record chart in a recorder. or a suitable storage register 11.

Equation 1 may be recognized as having three roots, each of 5Multipliers 65-68, along with subtractor 69 and adder 70,

which may have both a real component and an imaginary component The realcomponent is plotted relative to the zero axis 12 on the chart in therecorder of FIG. 2. The imaginary component is plotted relative to thezero axis 13. Each step in the two output signals from the system isshown on the chart and illustrates the manner in which the outputsignals migh approach the root values r r and r The system includes avoltage source which has a grounded output line 21, a positive outputline 22, and a negative output line 23. Five potentiometers 24-28 areconnected across lines 22 and 23 and have their center taps connected tothe grounded line 21. Source 20 is provided with a suitable indicatingmeter 29 and a suitable adjustment control so that voltages may beproduced on and applied to the potentiometers 24-28 in the magnitudedesired for any particular operation. 1

The arm on potentiometer 24 is connected to a bus 31. The arms onpotentiometers 25 and 27 are connected to terminals of a double-pole,double-throw switch 32. The output terminals of switch 32 are connectedto bus lines A and C, respectively.

In a similar manner, the arms of potentiometers 26 and 28 are connectedby way of a double-pole, double-throw switch 33 to bus lines B and D.The double-pole, double-throw switches are connected by way of linkage34 to a cam 35 driven by a motor 36. The switches 32 and 33 are thusperiodi' cally reversed so that when the switch arms are in their upperposition, the voltages from the arms on potentiometers 25, 26, 27, and28 are applied to lines A,;B, C, and D, respectively.

' When the switches 32 and 33 are in their lower position, the

voltages on the arms of potentiometers 25, 26, 27, and '28 are appliedto the bus lines C, D, A, and B, respectively. The voltages on lines Aand B represent a first estimate as to a first root of equation 1.Similarly, the voltages on lines C and D represent a second estimate.

That portion of F IG. 1 spanned by arrow 40 serves to evaluate equation1 for two initial complex arguments z, and Z2. The first complexargument z, is represented by the voltages on lines A and B. The secondcomplex argument Z2 is represented by the voltages on lines C and D. Theportion of FIG. 1 located on the left side of arrow 40 producesconditions at switches 41 and 42 representativeof the values of equation1 for the complex arguments 2, represented by the voltages on lines Aand B. Similarly, the system on the right side of arrow 40 producesvoltages at switches 43 and 44 representative of the solution ofequation. 1 for thecomplex arguments Z2 represented by the voltages onlines C and D.

That portion of the system spanned by arrow 45 serves to produce outputvoltages on output lines .46 and 47 which are representative of thesolution to equation'Z which represents a third approximation of thefirst root of equation 1. Thereafter, by iteration, signals aregenerated which represent the first root r of equation 1.

ln P16. 1, multiplier 51 is connected at both inputs to line A.Multiplier 52 is connected at both inputs to line B. Multiplier 53 isconnected at its inputs to lines A and B, respectively. Multiplier 54similarly is connected to lines A and B, respectively. Output lines frommultipliers 52 and 53 are connected to a subtractor 56. The output linesfrom multipliers 53 and 54 are connected to the inputs of an adder'57.The output line from subtractor 56 is connected to one input of adder58. The second input to adder 58 is connected to bus 31.

ROOT r For the values of the voltage placed on lines A-D, the voltage online 31 will be scaled to have a magnitude equal to 1. Thus, there isproduced on the output lines 59 and 60 voltages representative of .thefunction (2 1). Au adder 61 is conare then employed to carry out thecomplex multiplication of equation 1 to produce at switches 41 and 42voltages representative of the solution to equation l for the voltageson lines A and B. The multiplication and addition are carried out inaccordance with the relationships set forth in equations 7- l0.

In similar manner, the multipliers 81-84, the subtractor 86 and theadders 87, 88, 91, multipliers 95, 96, 97, and 98, subtractor'99, andadder cooperate to produce voltages at switches 43 and 44 which arerepresentative of the value of equation 1 for the particular voltagesapplied to the lines C and D.

Multipliers 101, 102, 103 and 104, along with subtractor 105 and adder106, produce voltages on lines 107 and 108 which are representative ofthe left side of the numerator of equation 2. Similarly, multipliers111-114, subtractor 115, and adder 116 produce voltages on output lines117 and 118 which are representative of the right side of the numeratorof equation 2.

More particularly, switch 41'is connected to one input of each ofmultipliers 101 and 104. Switch 42 is connected to one input of each ofmultipliers 102 and 103. In addition, line C is connected to one inputof each of multipliers 101 and 103. Line D is connected to one-input ofeach of multipliers 102 and 104. Thus, the product signals representingthe left side of the numerator of equation 2 areproduced on lines 107and 108. The voltage on line 107 represents the real part of theproduct, and the voltage on line 108 represents the complex part.Similarly, the voltage on line 117 represents the real part of the rightside of the numerator of equation 2, and the voltage on line 118represents the complex part. The subtraction indicated in the numeratorof equation 2 is performed by subtractors 119 and 120 in accordance withequation 8. The subtraction indicated in the denominator of equation 2is performed by subtractors 121 and 122. It will be noted that theswitches 43 and 44 are connected by way of lines 123 and 124 to inputson subtractors 121 and 122. Similarly, the voltages at switches 41 and42, as appearing on lines 125 and 126, are applied to the otherterminals of subtractors 121 and 122. Thus, the voltages on lines and131 represent the numerator of equation 2, and the voltages on lines132and 133 represent the denominator. The complex division is carried outby multipliers 134-139, adders and 141, subtractor 142, and dividers 143and 144 in accordance with the operations indicated in equation l0. Bythis means, there is produced on lines 46 and 47 voltages representativeof the value of equation 2 for the initial values of the voltagesappearing on lines A, B, C, and D.

With the system thus far described, assume that the switches 32 and 33are stationary in their upper position. ln such case, voltages will beproduced on lines 46 and 47 which are representative of a solution toequation 2 for the initial or starting voltages derived from thepotentiometers 24-28 and applied to lines A-D.

Under the control of motor 36, the operation is repeated for voltages onthe lines A-D which are changed stepwise until the change in the outputvoltages on lines 46 and 47 as between steps is within predeterminedlimits. The iterative process is accomplished by applying the voltageson lines 46 and 47 to storage units and then utilizingthe storedvoltages to adjust, through servolinkages, the positions of the taps onpotentiometers 25-28.

More particularly, line 46 is connected to a switch 150. Line 47 isconnected to a switch 151. Switches and 151 are actuated by linkage 34which leads to the cam 35 driven by motor 36. Switches 150 and 151 aresingle-pole, double-throw switches. One terminal of switch 150 isconnected by way of conductor 152 and switch 153 to a storage condenser154.

The second terminal is connected by way of switch 155 to a storagecondenser 156. One terminal of switch 151 is connected by way of switch157 to condenser 158, and the second terminal of switch 151 is connectedby way of switch 159 to storage condenser 160. The condensers 154, 156,158, and 160 are connected to a common ground terminal. Switches 153,155, 157 and 159 are single-pole, double-throw switches coupled togetherby way of linkage 161 to a solenoid 162. A switch 163 in the energizingcircuit from motor 36 is also actuated by linkage 161 to control theapplication of power to motor 36 from a suitable source 164. A manuallyoperated starting switch 165 is provided in series with the source 164and the motor 36.

Condenser 154 is connected by way of conductor 171 to one input of aservounit 175 which includes a servoamplifier and a servomotor having amechanical output linkage 179 shown dotted. Condenser 156 is connectedthrough conductor 172 to one input of a second servounit 176. Similarly,condensers 158 and 160 are connected through conductors 173 and 174 tothe inputs of servounits 177 and 178, respectively. The other inputterminals on the servounits l75-178 are supplied by the voltages onlines A-D, respectively. The servounit 175 serves to control theposition of the arm of potentiometer 25 by obtaining equality or balanceof the voltages at the servoinput in the manner which is well known.Servounit 176 serves to control the position of the arm of potentiometer27. Servc unit 177 controls the position of the arm of potentiometer 26and servounit 178 controls the position of the arm of potentiometer 28.

With the motor 36 energized, the voltage first appearing on lines 46 and47 will be stored on condensers 154 and 158. This output voltage is thenapplied to servounits 175 and 177 to adjust the positions of thepotentiometers 25 and 26. Simultaneously, the motor 36 reverses theposition of switches 32, 33, 150, and 151 so that the voltages frompotentiometers 27 and 28 now appear on lines A and B, and the voltagesfrom potentiometers 25 and 26, as adjusted by the servounits 175 and177, are applied to the lines C and D. The presence of new voltages onlines A-D causes new voltages to be produced on lines 46 and 47. The newvoltages are applied to condensers 156 and 160. The latter voltages thenactuate the servounits 176 and 178 to readjust the potentiometers 27 and28. While this is being done, the switches 32, 33, 150, and 151 areagain reversed so that successive values of voltages on lines 46 and 47are stored first on the condensers 154 and 158 and then on condensers156 and 160. Adjustments are made in the potentiometers 25- 28responsive to such voltages until such time as the difference betweenthe voltages on condensers 154 and 158 and the voltages on condensers156 and 160 are relatively small and are within predetermined limits.When the change falls within the limits set, the motor 36 isdeenergized. During the foregoing operation, the voltages on lines 46and 47 are stored and/or recorded as on the chart which shows the stepsleading to the root values r Motor 36 is energized and deenergized, inaccordance with one aspect of the present invention, by comparing thevoltages on condensers 158 and 160 and the voltages on condensers 154and 156. More particularly, a subtractor 180 is connected to condensers158 and 160 and a subtractor 181 is connected to condensers 154 and 156.An adder 182 is connected to the outputs of subtractors 180 and 181 andis of the type such that it will produce an output voltage which isequal to the absolute magnitude of the voltages on its two inputcircuits. The output voltage from adder 182 is applied to one input of asubtractor 183, the second input of which is supplied by a voltage froma battery 184. Battery 184 is adjusted to be equal to the voltagenecessary to cause the relay 162 to be energized. Relay 162 is connectedto the output of an amplifier 186 whose input is energized by a voltageequal to the difference between the voltage of battery 184 and thevoltage at the output of the adder 182. When a large difference voltageappears from either of the subtractors 180 and 181, there will be alarge voltage applied to the subtractor 183 so that the voltage of theoutput of amplifier 186 will be small compared to the voltage necessaryto energize the relay 162 as of polarity opposite to that required tooperate relay 162. However. when voltages appearing on lines 46 and 47are sufficiently close to a given root, then the relay 162 will beenergized under control of the voltage from the subtractor 183. Thisaction stops the first cycle of the system after having generatedvoltages which are representative of root r ROOTS r AND r,

In order to identify or evaluate a second root, it is then necessary torepeat the cycle operations just described but with the voltages fromthe outputs of units 69, 70, 99. and 100 adjusted as by operation ofthat portion of the system shown in FIG. 2.

That portion of the circuit shown in FIG. 2 is provided for successivelyeliminating the effect of each of the roots r,, r,, r, etc. from thesucceeding search for the next root. A keyto an understanding of theoperation of FIG. 2 and the cooperation thereof with FIG. 1 is expressedin equations 1 l l 4 and more fully depicted in table A, set out below.

It will be remembered that, starting with the voltages (a, ib,),equation 3, the voltages at the outputs of the units 69 and representthe evaluation of flz), unique for the given set of starting voltages.Similarly, starting with the voltages (a,

ib equation 4, the voltages produced at the outputs of units 99 and 100represent a second unique evaluation of f( z) for the given set ofstarting voltages. By applying the voltages at the outputs of units 69,70, 99, and 100 to the remaining portion of the circuitry of FIG. 1, athird estimate of the root r of equation 1 is produced in the form ofvoltages at output lines 46 and 47. The latter voltages are thensubstituted for the voltages on line C-D, while the initial voltages online C-D are substituted for the initial voltages on lines A and B byreversal of switches 32 and 33. The operations Z Z 2, .3, are thenfollowed until the root r is obtained. Motor 36 is at this timedeenergized and relay 162 is energized. Relay 162 'actuates switches 41,42, 43, and 44 to cause them to be latched into their right-handpositions. Thereafter, in finding the second root r the voltagesappearing at units 69, 70, 99, and 100 are modified in accordance withequations 1 l and 12, respectively. After root r, is generated, thegeneration of root r;, is initiated and in this sequence of operation,the voltages appearing at the outputs of units 69, 70, 99, and 100 aremodified in accordance with equations 13 and I4.

As indicated, the circuit of FIG. 2 serves primarily to effect themodification represented by equations 11 and l2 or by equations 13 and14.

More particularly, the voltages on lines A-D are each applied to aplurality of subtractor units. During the search for root r voltagesrepresentative of the value of f(z) for each of the two starting values2 and z are applied to FIG. 2 by way of conductors 201-204. The voltageson lines 201 and 202 are applied to one pair of inputs of a complexdivider 224. Voltages on lines 203 and 204'are applied to one pair ofinputs of a complex divider 225.

in FIG. 2, line A is connected to one input of each of subtractors 272,273, and 274. Line B similarly is connected to subtractors 282, 283, and284, line C, to subtractors 292, 293, and 294, and line D, tosubtractors 302, 303, and 304. The second terminals of subtractors 272and 292 are connected to the first terminal of stepping switch 251 andto storage condenser 260. The second terminals of subtractors 282 and302 are connected to the second terminal of stepping switch 252 g and tostorage condenser 261. The second terminals of sub tractors 273 and 293are connected to the third terminal of stepping switch 251 and tostorage condenser 264. Similarly,-.

switch 251 and to storage condenser 268, while the secondv to one pairof inputs of a complex multiplier 205. The second pair of inputs ofmultiplier 205 are connected to the outputs of subtractors 273 and 283.One output of multiplier 205 is connected to the third terminal ofstepping switch 253, while the other output of multiplier 205 isconnected to the third terminal of stepping switch 254. The two outputsof multiplier 205 are also connected to one pair of inputs of thecomplex multiplier 206, the second pair of inputs being supplied by theoutputs from subtractors 274 and 284. The outputs of multiplier 206 areconnected to the fourth terminal of stepping switches 253 and 254,respectively. Multipliers 207 and 208 correspond with multipliers 205and 206 and are connected to the circuits leading to the complex divider225.

The complex multipliers 205, 206, 207 and 208 will be of the typeillustrated in FIG. 1 and comprising multiplier elements 65-68,subtractor 69, and adder 70. They have been illustrated in simplifiedform in FIG. 2.

The arms on stepping switches 253 and 254 are connected by way ofconductors 210 and 211 to one pair of inputs of the complex divider 224.

The relay 162 drives six stepping switches 251-256. The steppingswitches 251 and 252 serve to apply the voltages representative of rootsas they appear on lines 216 and 217 to storage condensers 260 and 261,the latter voltages appearing on lines 262 and 263 leading from thesecond position terminals on switches 25] and 252. The voltagesrepresentative of the second root are stored on condensers 264 and 265,the latter voltages appearing on lines 266 and 267 leading from thethird terminals of switches 251 and 252. Voltages representative of thethird root are stored on condensers 268 and 269, the latter voltagesappearing on lines 270 and 271 leading from the stepping switches 251and 252.

Voltages representative of the initial values z appear on lines A and B.The voltages representative of the initial values 22 appear on lines Cand D. Three subtraction units 272-274 are provided for generatingvoltages representative of the real part of the modified voltages g h jof table A for runs 2, 3, and 4, respectively.

Lines 216 and 217 are connected by way of double-pole, single-throwswitch 219 to the arms of stepping switches 251 and 252. The switch 219is controlled by relay coil 162 so that voltages, representative of thefirst root r,, appear on lines 216 and 217 for storage on capacitors 260and 261 of voltages which are then applied to subtractors 272, 282, 292and 302.

The subtraction indicated in the denominator of equation I l is carriedout by subtractors 272 and 282. The subtraction indicated in thedenominator of equation 12 is carried out by subtractors 292 and 302.The division of equation 11 is carried out by the complex divider 224.The division of equation 12 is carried out by the complex divider 225.Thus, modified voltages appear at the output of the divider 224 with thevoltage representing the real part appearing on output line 226 and thevoltage representing the imaginary part appearing on output line 227.Modified voltages also appear at the output of divider 225 with thevoltage representing the real part appearing on line 228 and the voltagerepresenting the imaginary part appearing on line 229. Line 226 isconnected to the normally open terminal of latching switch 41. Line 227is connected to the normally open terminal of latching switch 42. Line228 is connected to the normally open terminal of latching switch 43,and line 229 is connected to the normally open terminal of latchingswitch 44. By energizing the relay 162, the switches 41-44 are latchedto maintain the connections to lines 226-229 so that the generation ofroot r will be independent of root r,.

When the motor 36 is deenergized after generating voltagesrepresentative of root r the subsequent energization of relay 162actuates switches 251-256, thereby producing a different set of voltageson lines 226-229. This will immediately cause different voltages toappear on lines 46 and 47. thus causing motor 36 to become energized toinitiate the generation of voltages leading to the third root r GENERALAPPLICATION It will now be apparent that the system of FIGS. I and 2 iscapable of handling polynomial functions generally. Only that portion ofFIG. 1 spanned by arrow 40 is peculiar to or singularly related to theparticular function being evaluated. This portion of the system will bemade more complex or less complex depending on the function to beevaluated so that there will be produced on output lines 41 and 42voltages representative of the real and complex parts of the functionfor a first starting voltage and voltages will be produced on lines 43and 44 representative of the same function for different startingvoltages.

The operation of that portion of the system shown in FIG. 2 serves toeliminate the effect on generation of successive roots of the existenceof roots in the system previously generated.

In the system of FIGS. 1 and 2, successive roots are generated inaccordance with the set of relationships set forth in table A.

TABLE A Voltages at From the foregoing, it will be seen that equation(E) indicates that in each case the function f(z) will be modified bydividing therein the product [(zr,)(Zr )(z-r,-)]. Equation (E) thussignifies the operation wherein all previously evaluated roots r,, r r,are employed in accordance with the relations of the equation (E) tomodify the function f(z). Such a set of voltages is indicated in column2 of table A.

In the circuit of FIG. 2, voltages are produced for application toswitches 4144 to permit the system to operate on the successivesequences of operations, as to produce voltages representative of a rootwhich is different than the root obtained in the preceding sequence.

It will be recognized that, while equation 1 has only three roots, thecircuit of FIG. 2 may be extended to accommodate functions having morethan three roots, specific circuitry being shown in FIG. 2. Additionalstages of subtraction and multiplication will be connected in tandem asneeded, in the manner represented by the subtractors 272274, andmultipliers 205 and 206, with the switches 253 and 254 to evaluate morethan four roots. FIG. 2 therefore illustrates the pattern that mayreadily be followed to add additional units.

STARTING CONDITIONS The starting voltages z,, Z z,, have been indicatedin table A as being the same for each series of operations leading toroots r,, r r r,,, respectively. In principle, the starting voltages maybe selected at random for each sequence. Further, the starting voltagesfor all sequences may be the same. Preferably, however, the voltages areselected as to be located in the region of the root to be generated. Forthis purpose, the switches 153, 155, 157, and 159 are connected to theirright-hand terminals prior to initiating each sequence. Lines 311-314extend to the circuit of FIG. 3 wherein starting values are generatedwhich are in the general vicinity of the root to be generated. Inaddition to lines 311-314, the line 31 extends to FIG. 3 along withlines 171 and 173. The starting voltages for generating root r, areproduced by unit 321. The starting voltages for generating root r aregenerated by unit 322; root r,,, by unit 323; root r,,, by unit 324; androot r by unit 325. Line 31 is connected to each of the units 321- 325to supply an input voltage of predetermined value. Output lines 326-330from units 321325 extend to a multiterminal stepping switch 331 which isdriven by linkage 161 extending from relay 162. The arm of the switch321 is connected by way of conductor 332 directly to line 311 and by wayofa voltage divider 333 to line 313. Line 312 is connected to ground.Line 314 is connected to line 332 by way of a voltage divider 334.

For the generation of the starting voltages for root r,, it will benotedthat line 31 is connected to the three inputs of an adder 336, theoutput of which is connected to a multiplier 337. The second inputofmultTplier 337 is connected to line 31. The output of multiplier 337 isconnected to a divider 338. the second terminal of which is connected toline 31. The output of divider 338 is connected to a subtractor unit339. Line 31 is also connected to the inputs ofa divider 340 whoseoutput is connected to the input ofthe unit 341 which generates avoltage corresponding with the nth root of the output voltage from thedivider 340. The output of the unit 341 is connected to the first inputof the subtractor unit 339. The output of unit 339 is connected by wayof output conductor 326 to the switch 331.

The operation of the system of FIG. 3 may best be understood byreferring to the following relationships and the diagram of FIG. 4. Itis known that a preferred set of voltages to start the generation of thefirst root r, of a given polynomial, such as the polynomial of equation1, depends upon the order of the polynomial and upon certain of thecoefficients therein. In the polynomial of equation I, i.e.,f(z) z z zl, a quantity known as the centroid C,, as defined in equation I5, isthe centroid of the system with the first root present. The geometricmean radius of the roots relative to the centroid is expressed inequation 16. In the operation of FIG. 3, a voltage representative of thecentroid C appears at the output of the divider unit 338. The voltagerepresentative of the radius R, appears at the output unit 341. For theexample of equation I, the plot of FIG. 4 is representative of thelatter voltages. More particularly, the output of the unit 338,corresponding with the centroid C,, would have a value of (l/3). Thus,the centroid C, has been plotted in the complex plane of FIG. 4 at (l/3i0). The radius R, has also been plotted so that the circle 345represents t l '1e locus of radius R,.

is applied by line 332 to conductor 311. This voltage represents a,.Since the first starting point lies on the x-axis,

the voltage ib, is of zero magnitude. Therefore, conductor 312 r isconnected to ground. The segment K of FIG. 4 represents about l/lO of aradian and thus the resistances 333 form a voltage divider that appliesto conductor 313 a voltage slightly less than the radius R,. Resistors334 form a voltage divider which applies to conductor 314 the voltagewhich is about l/lO ofthe radius R, and thus equal to the segment K.

Equation I will be found to have roots at (l (+1), and (i) as indicatedin FIG. 4. Thus, starting with the first starting values plotted in FIG.4, root r, will first be located. Thereafter, the voltages representingthe root r, will be stored in unit 322 in the same manner in which theroot voltages are stored, in FIG. 2, on condensers 260 and 261.Additionally, voltages of radius R, and'centroid C, are stored in unit322 so that a voltage will be produced and applied to conductor 327, andthence by way of switch 331, to conductor 332 to produce on conductors311-314 starting voltages for the second root The relationship betweenthe voltages produced in the system of FIG. 3 may be better understoodby referring to the relationships illustrated in equations l526 whichidentify the successive starting voltages. In each case, with the relay162 of FIG. 1 energized, new voltages are applied to condensers 154,156,158, and which cause the relay 162 to be deenergized, thereby startingthe search for the next root.

The operations of FIG. 3 are represented mathematically in equationsl526. To start the generation of voltages representative of root r,,voltages representative of a first centroid C, and a first radius R, aregenerated in accordance with the following relationship:

With the voltages representative of centroid C, and radius R, availableat the outputs of units 341 and 338, the starting voltages applied toconductors 311-314 are produced in accordance with the followingrelationships:

and

More accurately, the voltage would be slightly less than a, and would beequal to R, Cos 0. This voltage, for all practical purposes, would beabout equal to R, since 0 is 1/10 radians. From the starting voltages onlines 311-314, the system of FIG. 1 is set into operation to generatevoltages representative of root r,. With voltages representative of rootr, stored on condensers 154, 156, 158, and 160, the operation of thesystem of FIG. 1 terminated, and unit 322 of FIG. 3 then generatesvoltages representative of the centroid C and radius R in accordancewith the following relationships:

With the centroid and radius voltages C and R known, new startingvoltages are generated in accordance with the following relationships:

and

From the latter starting voltages, the system of FIG. 1 generatesvoltages representative of root r Thereafter, centroid voltages C andradius voltages R are produced in unit 323 in accordance with thefollowing relationships:

3 n-2 (23) and L R: not) H a a '2)( a 1)- With the centroid and radiusvoltages C and R known, new starting voltages are generated inaccordance with the follow ing relationships:

and

From the starting voltages of equations 25 and 26, the system of FIG. 1then proceeds to generate voltages representative of root r It will beapparent that the successive starting voltages may be evaluatedindependently of the system of FIG. 3 and that voltages may, at least inpart, be manually applied to and stored on condensers 154, 156, 158 and160 between successive sequences of root voltage generating operations.For example, in FIG. 3 unit 321 may be greatly simplified where thefunction is equation I having coefficients A A,, A,,, all numericallyequal to one. It will immediately be apparent that the first centroidwill have a value C, (l/3 +i0) and that the radius R, will be equal toone. This would permit selection of the voltages to be stored on thestorage condensers of F IG. I quite independently of the operation ofunit 321 of FIG. 3. A voltage applied to line 332 would result, by meansof the connections including dividers 333 and 334 to lines 311-314, inhighly desirable starting voltages for FIG. 1. The centroid C and theradius R would differ from the corresponding values C, and R, asindicated in equations 19 and 20. The latter parameters may bedetermined and starting voltages selected for generation of root r Thus,it will be understood that starting voltages may be selected at random;may be the same for all sequences; may be computed and introduced intothe system of FIG. 1 by manual operation; or may be generatedautomatically through the operation of a system of a type indicated inFIG. 3.

POLES OF EQUATION l The systems shown in FIGS. 1-3 may be employed asabove described to identify or generate signals which represent theroots of a function such as equation I. The same circuit may be employedto identify poles by merely eliminating FIG. 3 and changing connectionsin FIG. 1 between lines A, B, C, and D and lines 121, 122, and 126.Further. changes are made at connections and at the input lines tosubtractors 121 and 122. This portion of the circuit of FIG. 1, with thechanges above noted, is illustrated in FIG. 5 wherein the same referencecharacters are employed as in FIG. 1.

With the circuit of FIG. I thus modified, the operation proceeds inaccordance with the following equation:

nor-fem 7) It will be recognized that, in the generation of voltagesrepresentative of poles, at far more stringent requirement is placedupon the components of the system of FIG. 1 than in generation of roots,since the voltages representing the functions flz),,- and f(z),., ofequation 27, for example, will become very large at the poles, thusrequiring a wide dynamic range in the system. In an analogue system ofthe type disclosed, poles may be located by symmetry with respect tovoltages which reach the limits of the dynamic range of the analoguesystem. To the extent that the analogue system lacks dynamic range togenerate voltages representative of the poles to their precise oraccurate pole point, the operation will be limited to an approximationof the poles unless an analytical representation of the inverse of thefunction can be found which does not require excessive range for thedenominator of the resulting function SYSTEM COMPONENTS The systems ofFIGS. 1 and 2 show schematic representations of circuit components whichare well known in the art and for this reason are not shown in detail.The potentiometers 24-28 may be decade-type variable potentiometers ofconstruction and the desired range to provide the voltages for theoperation of the system with the accuracy desired. For example, thesubtractors may be of the type described in Waveforms, RadiationLaboratory Series, Volume 93, Mc- Graw-I-Iill (1947) in the sectionentitled Addition and Subtraction of Voltages and Currents, page 629. Asuitable subtractor is described at page 642 under the heading CathodeCoupling." Adders are also described in the same section. A suitableadder is shown in FIG. 18.1 at page 631 of the above reference. Themultipliers and dividers may be as described in the section entitledMultiplication and Division" in the above-identified reference.Multipliers such as employed in the circuit I9.l of the reference may beof the type disclosed in US. Pat. No. 2,982,942 to J.E. White. A directlinkage has been indicated from the motor 36 to the drive of recorder10. It is to be understood that other drive means, suchas wellknownSelsyn systems, may be employed to coordinate or synchronize the motionof the recorder chart with the operation of motor 36. Servounits l75I78,in general, are well known and may be of the type illustrated anddescribed in Handbook of Automation Computation and Control, by Grabbeet al. John Wiley & Sons, Inc. (1961), page 23.08, and as further andmore generally explained in the section of the same work beginning atpage 6.23.

The unit 341 of FIG. 3, for generating an output signal representativeof the root of the input signal may be of a wellknown type, such as thatdescribed in Electronic Analogue Computers, by Korn and Korn,McGraw-Hill (I956), and

more specifically, as illustrated in table 5, No. 5.5 at page 425.

DIGITAL MODE The description of FIGS. 1 and 2 has dealt primarily withthe use of an analogue system for carrying out the method of the presentinvention. It will be understood that the method may be practicedthrough use of systems other than analogues of the type above describedin detail; More particularly, each operation embodied in the productionof root voltages in FIGS. l-3 and in FIG. 5 may be carried out throughthe use of digital computers, which for many operations will be found tobe more flexible and versatile. The method, therefore, is not limited tothe particular form of analogue apparatus disclosed herein. The analoguesystem will be found to be helpful in understanding the steps involvedand the relationships of the various quantities or conditions produced.On the other hand, a digital system will in general be preferred for usein carrying out the present invention. It is to be emphasized that themethod is not limited to the production of the particular roots ofequation I as above described, but that equation I was chosen merely asan example ofa low order problem by which the invention might beexplained as applied to generation of conditions which represent complexroots of a given signal.

In carrying out the method of the present invention on general purposedigital computers; the operating system" effectively is built throughinstructions to the computer by way of a program. While computerprograms having a specific objective differ in specific codes or in themanner of expression, there are a number of well known and widelyrecognized codes routinely used by those skilled in the art. A computerprogram expressed in one such code for carrying out the presentinvention on a general purpose computer is set forth in the followingtables B-E. The main program appears in table B and is expressed in thefamiliar computer program language FORTRAN-IV.

Table B CALL FUN (C,D,U,V,Y0,Y I ,I,N) CALL .IONSTR (M,C,D,D I,D2,I.I,U,V,Ul,V l ,IPP) 10 CALL FUN (C,D,U,V,Y0,Y l ,IPP,N) l ICONTINUE X=UIU Y=VIV CALL REDUCE (DI ,D2,YO,Y, l ,U,V,I.I)

9 CONTINUE CALL FUN (C,D,U1 ,V,Y2,Y3,IPP,N) CALL REDUCE (DI,D2,Y2,Y3,UI,V,IJ) XD =Y0Y2"(Table B, continued) YD=Y1Y3 IF (XD.EQ.0.) GO TO 19CT=YD/XD CD=XD+CT*YD CR=(X+CT*Y)/CD CI=(Y-CT*X)/CD 2l CONTINUEX=Y2*CRY3*CI Y=Y2*CI+Y3*CR CWl =UI+X CW2 =Vl+Y L=L+ I IF (L.GT.50)GO TO22 TEST =ABS(X) (ABS(CWI )+l .E20)*ACC TEST 2=ABS(Y (ABS(CW2)-lE-20)*ACC IF (TEST.LT.O. .OR.TEST 2.LT.0.) GO TO 8 ICNT=0 12 CONTINUEUI =CWI VI =CW2 Y0=Y2 Yl =Y3 GO TO 9 40 CW2 =0. (Table B, continued) IF(TEST.GT.O.) GO TO I2 GO TO 39 I 38 CWI =0.

IF (TEST 2.GT.O.) GO TO 12 GO TO 39 8 ART=ABS(CS2/CWI) IF (ART.LT.1.EI2)GO TO 40 IF (ART.GT.I.EI2) GO TO 38 IF (TEST.LT.O. .AND.TEST 2.LT.0.) GOTO 29 ICNT =0 GO TO 12 29 CONTINUE ICNT=ICNT+I IF (ICNT.LT.2) GO TO 1239 CONTINUE LX =0 U=IJ+I DI(I.I)=CWI D2(IJ)=CW2 IF(IJ.GE.M)RETURN CALL.IONSTR (M,C,D,D I ,D2,l.l,U,V,U l ,V I ,IPP) 23 CONTINUE L=l GO TO 10(Table B, continued) 13 ITT=IJ+I WRITE (6,I06)I'IT,CWI ,CW2,X,Y LE=-lRETURN 19 CONTINUE IF (YD.EQ.O.) GO TO 20 CR=Y/YD CI=X/YD GO TO 21 20CONTINUE GO TO 39 22 CONTINUE LX=LX+1 lF (LX.GT.20) GO TO 13 CALL.IONSTR (M,C,D,DI,D2,IJ,U,V,U I ,V I ,5) GO TO 23 121 LE=1 Thesubroutine called for in table B to locate, from a given WRITE (6,1 l IM polynomial, an estimate of the location in the complex plane RETURN ofwhere the roots are located is set forth in table D.

I02 FORMAT (24H RESPONSE DIFF. ZERO.//)

106 FORMAT (17H ERRORFAILED ON,l5,36l-I 5 TableD RooTIAsT VALUE ANDINCREMENT MBFTC JONS 1 ARE SUBROUTINE JONSTR M,c,D,D1.D2.II.U,v.U1 v 111 FORMAT (22H ERROR. POLYDEOREE 15/) 1,100)

END

DIMENSION C( I ),D( l ),Dl( 1 ),D2( l) DOUBLE PRECISION U,V,U l,V1,XC,YC,X,Y GOTO(1,2,3,3,4),IGO 1 CONTINUE N== M+l DO 8 I= 1,N IF(C(I).NE.0.) GO TO 9 8 U I] l 9 FN= M-LI RR= ABS(C(N)) (Table D,continued) RR=RR**(1./FN) YC=O. XC=-C(M)/(FN*C(N))+ l.0E-6 5 CONTINUE IF(IJ.EQ.0) GO TO 21 DO I= l,I.I D2 (I)=O. 20 D1 (I) 0. 21 CONTINUE In theprogram of table B, the following relationships will be 10 understood tobe effective:

where N=M+1. v 25 b. IflCM =0, all D(i) are equal to 0. c. If LE isgreater than zero, the program has Operated Without error. d. If LE isless than zero, error is present. XM e. By way of example, consider thecommand Call 6 CONTINUE UNIVR (5,C,D,D l ,D2,0,LE). This command CALLFUN (QDYXCYCXYJDOM implies that there is a polynomial of degree 5 CALLREDUCE (DLDLXXXQYQU) with only real coefficients which are contained 5/in an array labeled C. D is now a dummy array. R= R/ R 1 0E-6 Roots ofthe polynomial are returned to the ex- C+ R ecutive program in DI and D2arrays for real =YC and imaginary parts. Tests would be made on LE 7CONTI to see whether the executive program should be XA= 2*XM permittedto continue.

VI=YC+R*(SIN(XA) In the same format, the subroutine called for in tableB for evaluating a polynomial of the general form ax" -l-ax" an RETURN=0 is set forth in table C. 2 CONTINUE (Table D, continued) TABLECDOI8I=1,N

The subroutine called for in table B for reducing or eliminating theeffect of previously generated root functions is set forth in table E.

Table E $IBFF C REDS SUBROUTINE REDUCE (D l ,D2,A,B,U,V,l.l)

DOUBLE PRECISION A,B.U,V,X,Y,W,Z,R DIMENSION Dl( 1 ),D2( 1) IF(I.I.EQ.O) GO TO 2 DO I I= 1,11 X= UDI(I) Y= VD2(I) IF (X.EQ.0.) GO TO 3R= Y/X Y= X+R*Y X= I. /Y Y=R/Y GO TO 4 3 CONTINUE IF (Y.EQ.O.)Y= l.E-l5Y=l./Y 4 CONTINUE Z=A*XB*Y W= A*Y+B*X A= Z B= W (Table E, continued) 1CONTINUE 2 CONTINUE RETURN END ANTENNA PATTERN SYNTHESIS The methodabove described has been applied to the problem of radar antennasynthesis. As described in Introduction to Radar Systems, by Skolnik,McGraw-Hill, 1962, beginning at page 320, it is known that an antennaradiation pattern of N elements may be expressed as a polynomial ofdegree N-l. A given radiation pattern such as shown at page 322, FIG.7.50a, may represent a radiation pattern for the far field of amultielement antenna. The Fourier analysis of a desired pattern may thenbe expressed in terms of a polynomial. Generation of roots of suchpolynomials, in accordance with the present method, permits thedetermination of the phase and amplitude of the excitation current foreach of the elements of the antenna array.

The method has been applied to this and to other special cases for whichother methods exist, and has been found to compare extremely favorablywith older techniques for handling such cases. The method has also beenapplied to operations involving sets of linear equations, as describedbelow.

SETS OF LINEAR EQUATIONS It is known that the condition that the set oflinear equations possesses nonnull solutions is equivalent to thedetermination of those values of A for which the determinant ofvanishes. This restatement of the problem does not always lead directlyto numerical analysis using classical methods on the expanded equation.Two problems present themselves. First, if the order, n, of the matrixis moderately large, an analytical expansion of the determinant islaborious and time consuming, involving of the order of n algebraicoperations. Secondly, even in the event that the functional expansion ofthe determinant is available, either analytically or numerically, thedetermination of the zeros of this function is generally a difficultmatter in itself, as in the problem of equation C, above. This isparticularly true if N is large or if depends on A in any but the mostelementary fashion. A problem of finding eigenvalues of even aIOth-order matrix whose elements consist of transcendental functions orpolynomials in A would present an almost insurmountable task, bothanalytically and computationally, if prior procedures were used.

The present invention provides a simple solution as the followingrelationships show. Consider the set of equations HX =Y (2s) whereY=(y1,y2, 11/ 0 is given. Cramers well-known rule states that isproportional to (det H);

thus some component of will consequently become unbounded as Aapproaches one of its eigenvalues. Naming the eigenvalues A A etc. andassuming they are distinct, at least one of the components, for exampleX may be defined as:

where the numbers, C, are constants (dependent upon and Y).

For values of A sufficiently close to one of the eigenvalues, say A afurther approximation is:

XIN Cu This approximation forms the basis of the procedure which is nowdescribed:

This requires of order 0(n operations by reduction methods.

3. Focusing attention upon one of the components of X, say x makethe'approximations N 11 N o X1(z)1-- and X (z) M (33) which upondivision yield a new estimate of A 1 X1( 1X (z) 34 4. Continue thisprocess to generate the sequence of estimates according to the recursiverelation 5. The sequence converges to an eigenvalue, i.e., to the polesof equation 29 which are zeros of the determinant. This eigenvalue maythen be removed by formation of a new function X (z) F X, (z)(A A andthe next eigenvalue found efiibib ing X,. In this manner all theeigenvalues are found in.sequence.

Roots of a wide variety of functions characterized by equation A havebeen generated using the FORTRAN program of tables B--E for the IBM 7040computer. These functions have ranged from a 46th-degreecomplex-coefficient polynomial, the roots of which were extracted to 7decimal digit accuracy (using 36-bit words), to the meromorphic functionsech [(2) At], the first l0 roots of which were sought and generatedaccurate to 8 digits. The accuracy to which roots may be" generated islimited by the ability to compute the value of the function over therange of the roots. The removal of the effects of roots by redefiningthe function as specified in equations 1 l- 14 reduces the effects oflarge differences in magnitudes of roots which is an advantage overother techniques, and effects of the order in which roots are foundwhich also is a decided improvement over prior art methods.

Because the eigenvalue operation and the general rootfinding operationare so common in their nature, the examples which follow relate only toeigenvalue operations The method has been applied with success to avariety of problems in which the coefficients of the array werenonlinear functions of Example 2 Eberlein B Present method theeigenvalue. In order to compare the results to previously 10 Example 3tested pathological cases, the following examples are of the specialform Eberlein B Present method 2 0 0 0 2. 008 2. 0000000 l 2 o o 1.9921.9999999 l5 0 1 2 0 2. 00s 2. 9090999 0 0 1 2 1.991 2.0000000 Example 4Lotkin Eberlein B Present method 1 1 1 1 1 1 2.132376 2.132376 2.1323763V. K! 14 93 $6 $6 .2214068 .2214066 .2214068 1-9 M .3184330(1).3184328(1) .31843305(1) V4 93 .8983233(-3) .8983258(3) .89832695(3) $656 1706278(4) 1706200(4) 17063663(-4) $6 5% ts $6 Mo Mi .1394499(6).1443702(6) -.145928(6) Because the present method is completelygeneral, the true test for special cases lies in the accuracy of theroots found Exam le 5 rather than speed. This is tru; lziecauig,forgarge matrices of Eberlein B Present method special form, the presentmet 0 cou not ope to compete,

. 5+1 1 5. 011 9 timewise, with prior methods which may and do make useof O 00 9 4 9999995 00000031 h ses however partlcular k-nqv-vledge- Evenm sue ca 5+1 4.99988+1.0016l 4.9999999+.99999990I reasonably closeinitial estimates should tend to balance the scales somewhat. Forrelatively small matrices, the time dif- 40 SH 1 5.000115%99871ammoolkggggggw ference is quite small, even with poor estimates. In allof the 0 following examples, reduction methods using 36-bit words 5H 4'99984 +1. 0013i 5 0000001 +1. 0000000 were employed to computedeterminant values.

Example 6 True Eberlein B Present method 15 11 6 9 15 1 .9999 1-W 1 3 93 s 1.50016+3.57064i 1. 505+3. 57i 1. 4984986+3. 5702653 7 6 6 3 11 1.50016-1-3. 57064i 1. 495+3. 571 1. 5014906+3. 571103 7 7 5 3 -111.500163.57064i 1. 505-3. mi 1. 4999583-3. 5712705} 17 i2 5 -10 16 1.50016-3. 570M 1. 495-45. 57i 1. 4986422-3. 57022011 Example I a -l-=(i+l )(N+S), a,, =i(N-i+l ll ir+i ir i )i a =0,otherwise. 60 For allattempted cases including cases well known to be The choices S -6.5, N 6create a situation of a pair of eigenvectors essentially parallel andlead to the following results, which are compared with prior art methodsas indicated in the following tables, and specifically, with the methodsdisclosed by Eberlein and by Lotkin, as hereinafter set forth.

pathological and presenting difficulties to other methods, the methoddescribed herein possesses great advantages.

The method and system above described accommodate electrical signals orother conditions which may represent a wide variety of sets of data.Only that part of the system designated by arrow 40, FIG. 1, is uniqueto the particular set. The remainder of the system of FIGS. 1 and 2forms a loop having two input channels 41, 42 and 43, 44, and two outputchannels A, B, and C, D. The loop is adapted to be closed by thefunction evaluating unit designated by arrow 40. The control means inFIG. 1 includes motor 36, and the components driven thereby, serve tocause generation of successive iterative conditions, each of suchconditions representing complex numbers. The loop has structure dictatedby or conforming with the. properties of equation 2. As expressed inequation 2,

the loop will include units designated by arrow 40. The control meansincludes the comparison means 180-186, for comparing each system outputcondition with the next preceding output condition and for causing astop, by actuation of relay 162, in the operation of the system when thechange between successive system output conditions falls within certainlimits. The control unit further includes means for restarting thesystem to run to succeeding stops. That portion of the system shown inFIG. 2 cooperates with the system of FIG. 1 to modify all outputconditions from the function evaluating unit which are produced afterthe first stop in dependence upon the magnitude of each of the systemoutput conditions as stored on condensers 154, 156, 158, and 160, nextpreceding each stop. The latter system output conditions are stored oncondensers 260-261, 264-265, and 269-270, as representative ofroots r rr The system may be constructed in analogue form as shown in FIGS. 1-3,or it may be constructed in response to a computer program as set out intables B-E. In either system, physical conditions, for example voltages,are produced to represent functions which are readily described in theshorthand of a mathematician but, nonetheless, accurately described asin the equations employed in the foregoing description and translatableinto structure as also described herein.

In the above examples 1-6 the comparisons are made with publishedresults using prior art methods. Eberlein A is more fully detailed inMathematics of Computations, Volume 18, page 296 (I964). Eberlein B ismore fully detailed in The Journal of The Society of Industrial AppliedMathematics, Volume l0, page 74, (I962). Lotkin is more fully detailedin the Quarterly of Applied Mathematics, Volume l4, page 267 1956).

From the foregoing, it will be seen that the present method is ofgeneral utility in generating conditions representative of the complexroots of a function where the function is of such nature that it variesin dependence upon a single complex variable condition. A pair ofcomplex root conditions are first stored to represent differentestimates of a first root of the function. Two different functionconditions are generated to represent the value of the function for eachof the two stored conditions. The members of the pair of stored rootconditions are then alternately replaced with successive new rootconditions which are produced by subtracting from the product of thesecond of the root conditions and the first of the function conditions,the product of the first of the root conditions and the second of thefunction conditions, and then dividing the difference by the differencebetween the second of the function conditions subtracted from the firstof the function conditions. The replacement continues until thedifference between the penultimate and end stored root conditions iswithin a predetermined limit. The end root condition is separatelystored. Thereafter, additional end root conditions are generated, witheach function condition generated subsequent to the generation of thefirst end root condition being modified in dependence upon allpreviously generated end root conditions.

Further, the method may be described in terms of sets of conditions orvoltages where a complex function f(z) is involved and electricalsignals representative of zeros of such function are to be generated. Afirst sets of electrical signals may represent a first estimate z, ofthe first of the zeros. A set 3, of electrical signals may represent asecond estimate Z2 of the first of the zeros. In response to a group ofsignals representing the function f(Z) and to the set S, of signals, aset Sm of electrical signals, representative of f(z) is generated.Similarly, in response to the group of signals and to the set 8, ofsignals, a set Sm of electrical signals, representative of f (z) 2 isgenerated v In response to fhe sets S Sq, Sm Sm f l ri l signals, a setS of electrical signals, representative of an estimate is generated,i.e.:

A sequence of setsS S, S, S of electrical signals, representative ofestimates z,,..., z,, z,,,,:,, of the first zero, is generated whereinthe set Si of signals is generated in response 108 to S to the set ofelectrical signals representative off(z),.., nd to the set of electricalsignals representative of f(z),. i.e.:

In response to the sets S andS,,, of signals, an electrical condition isgenerated, representative of the existence of a change between the setS,,, and the set 5,, which is within predetermined limits. Thedifference between successive root conditions is compared to apredetermined limiting or error signal. When the difference is equal toor less than said error signal, the generation of a further rootcondition is terminated and a new routine is started. In response to theelectrical condition produced only when the change is withinpredetermined limits, the generation of the sequence is terminated.Thereafter, in response to the group of electrical signals and to theset S a modified group of electrical signals, representative of [f( z-zis generated and applied in the aforementioned processing steps to themodified group of signals to generate a set of electrical signals,representative of a first zero of and hence representative of a secondzero of f (z).

The system, whether in digital or analogue form, will include means forstoring a pair of complex root conditions which represent differentestimates of a first root of said function and means for generating twodifferent function conditions representative of the value of thefunction for each of the stored conditions. Means are provided forgenerating successive new root conditions and including means forsubtracting from the product of thesecond of the root conditions and thefirst of the function conditions, the product of the first of the rootconditions and the second of the function conditions, and for dividingthe difference by the difference between the second of the functionconditions subtracted from the first of the function conditions. Afeedback system alternately replaces the members of the pair of rootconditions with the successively produced new root conditions.

Means are provided for comparing each new root condition with apreceding root condition, and are operative, when the difference betweenthe penultimate and end stored root conditions is within a predeterminedlimit, to interrupt generation of new root conditions. Means areprovided for separately storing each end root condition.

Finally, means are provided for initiating subsequent generation of newroot conditions leading to additional end root conditions, includingmeans for modifying each function condition generated subsequent to thegeneration of the first end root condition in dependence upon allpreviously generated end root conditions.

Having described the invention in connection with certain specificembodiments thereof, it is to be understood that further modificationsmay now suggest themselves to those skilled in the art and it isintended to cover such modifications as fall within the scope of theappended claims.

We claim:

1. The method of generating physical conditions representing complexroots ofa function f(:) ofa single complex variable quantity 1 whichcomprises:

a. generating a first sequence of physical conditions within a computereach representing approximations of a first of said roots according tothe recursive relation where n is the index ofthe conditions in saidsequence,

b. terminating said sequence in a first root condition, and

c. successively generating in said computer subsequent sequences of saidconditions to produce the second, third. nth root conditions, where inthe case of each sequence leading to a root condition, prior to eachsequence the function f is modified by dividing the conditionrepresentative thereof by a condition representative of the product ofthe differences between the condition upon which the function f (z)depends and the previously stored end root conditions, namely,

LU -n) i= l, 2,3,4, Ir.

2. The method of generating conditions representative of the complexroots of a function, which function varies in dependence upon a singlecomplex variable condition, which comprises:

a. inputting to a computer a pair of physically represented complex rootconditions which represent different estimates of a first root of saidfunction,

b. generating in said computer different physical conditionsrepresentative ofthe value of said function for each of the storedconditions.

c. alternately replacing said complex root condition with successiveroot conditions produced by subtracting from the product of the secondof said root conditions and the first of said function conditions theproduct of the first of said root conditions and the second of saidfunction conditions, and dividing the difference by the differencebetween the second of said function conditions subtracted from the firstof said function conditions, until the difference between thepenultimate and end stored root conditions is below a predeterminedlimit, separately storing the end root condition, and similarlygenerating in said computer additional end root conditions with eachsaid function condition generated subsequent to the generation of thefirst end root condition modified by dividing each function condition bya condition representative of the product of the differences between thecondition upon which the function condition depends and previouslystored end root conditions.

3. The method of claim 2 in which each starting pair of root conditionsis representative of points spaced from the centroid of all unknown rootconditions a distance of the order of the mean radius of said unknownroot conditions.

4. The method of claim 2 in which each starting pair of root conditionsis generated in dependence upon the coefficients of elements of saidfunction closely to approximate end root conditions as startingconditions.

5. The method of claim 2 in which the root conditions of said pair aregenerated to have values in the region of the centroid ofall of theroots of said function.

6. The method of generating a pair of voltages representative of thecomplex roots of a function, which function varies in dependence upon asingle complex variable voltage, which comprises:

a. inputting to a computer a pair of complex root voltages whichrepresent different estimates of a first root of said function,

b. generating in said computer different function voltagesrepresentative of the value of said function for each of said rootvoltages,

. alternately replacing said root voltages with successive root voltagesproduced by subtracting from the product of the second of said rootvoltages and the first of said function voltages, the product of thefirst of said root voltages and the second of said function voltages,and dividing the difference by the difference between the second of saidfunction voltages subtracted from the first of said function voltages,until the difference between the penultimate and end stored rootvoltages is sufficiently small,

. separately storing the end root voltage, and

e. thereafter generating additional end root voltages with each saidfunction voltage generated subsequent to the generation of the first endroot voltage modified by dividing said single complex variable voltageby a voltage representative of the product of the differences betweenthe condition upon which said voltage depends and the previously storedroot voltages.

7. The method of generating conditions representative of the complexroots ofa function, which function varies in dependence upon a singlecomplex variable condition, which comprises:

a. inputting to a computer a pair of complex root conditions whichrepresent different estimates of a first root of said function,

b. generating in said computer different function conditionsrepresentative of the value of said function for each of the rootconditions,

. alternately replacing the members of said pair with successive rootconditions produced by subtracting from the product of the second ofsaid root conditions and the first of said function conditions, theproduct of the first of said root conditions and the second of saidfunction conditions, and dividing the difference by the differencebetween the second of said function conditions subtracted from the firstof said function conditions, until the difference between thepenultimate and end stored root conditions is within predeterminedlimits,

d. separately storing the end root condition, and

e. thereafter generating additional end root conditions with each saidfunction condition generated subsequent to the generation of the firstend root condition modified by dividing said single complex variablecondition by a condition representative of the product of thedifferences between the condition upon which said single complexvariable condition depends and the previously stored root conditions.

8. The method of generating conditions representative of the complexpoles of a function, which function varies in dependence upon a singlecomplex variable condition, which comprises:

a. inputting to a computer a pair of complex pole conditions whichrepresent different estimates of a first pole of said function,

b. generating in said computer different function conditionsrepresentative of the value of said function for each of the rootconditions,

. alternately replacing the members of said pair with successive poleconditions produced by subtracting from the product of the second ofsaid pole conditions and the second of said function conditions, theproduct of the first of said pole conditions and the first of saidfunction conditions, and dividing the difference by the differencebetween the first of said function conditions subtracted fromithe secondof said function conditions, until the difference between thepenultimate and end stored pole conditions is sufficiently small,

. separately storing the end pole condition, and

. thereafter generating additional end pole conditions with each saidfunction condition generated subsequent to the generation of the firstend pole condition modified by representing approximations of a first ofsaid roots acl cording to the recursive relation 2 n-Ij )nl nlf( )n2 nj( )n-l f( )n-2 where n is the index of conditions in said sequence,

b. terminating said sequence in a first root condition, and

c. generating subsequent sequences of said conditions to produce thesecond, third, nth root conditions, where in the case of each sequenceleading to a root condition, and in number corresponding with the orderof said function f (2) with elimination in each said subsequent sequenceof the effect of all earlier generated root conditions by dividing thecondition representative of f (z) by a condition representative of theproduct of the difference between said condition upon which f (z)depends and the previously stored end root conditions.

10. The method of generating physical conditions representing complexroots of a function f (z) of a single complex variable quantity z whichcomprises:

a. generating a first sequence of physical conditions each representingapproximations of a first of said roots according to the recursiverelation where n is the index of conditions in said sequence,

b. terminating said sequence in a first root condition, and

c. successively generating subsequent sequences of said conditions toproduce the second, third, nth root conditions, where in the case ofeach sequence leading to a root condition, with modification in eachsaid subsequent sequence of the conditions representative of the quantities f (z),,-- and f (z),, by dividing the condition representative ofthe quantity f (z),, by the product l(zl and by dividing f (z),, by acondition representative of the quantity 2(z-2) to eliminate the effectof all earlier generated root conditions.

11. The method of producing physical representations of the roots of acomplex function f (z) which comprises:

a. generating a first set of starting conditions z representative of afirst estimate of the location of a first root,

b. generating a second set of starting conditions Z2 representative of asecond estimate of the location of said first root,

c. generating a third set of starting conditions Z3 representative of athird and improved description of the location of said first rootaccording to the relation where n =3,

d. thereafter substituting members of the series Z Z for representationsof z, and 2 in the sequence Z3 for z,, z,

for Z2, Z5 for 2 until the change between conditions representing 2;,and 2,, is within the predetermined limits whereby conditions 1,,represent said first root, and

e. repeating the aforementioned steps with successive modification ofthe conditions representative of the functions f(z) f(z),. themodification being in accordance with the relation i= 1, 2, 3, 4, n toeliminate the effect therein of root functions previously generated. n

12. The method of processing a group of electrical signalsrepresentative of values of a complex function f (z) to produceelectrical signals representative of "the value of roots of said 0function which comprises:

a. generating a set S of electrical signals. representative of a firstestimate 2:, for the first of said roots,

b. generating a set S, of electrical signals, representative of a secondestimate Z for said first of said roots,

c. generating, in response to said group of signals and to said set S,of signals, a set S of electrical signals, representative ofa value f(2) d. generating, in response to said group of signals and to said setS, of signals, a set Sflz)2 of electrical signals, representativeoff(z)ae. generating, in response to said set S 5 S S of electricalsignals, a set S of electrical signals, representative of an estimate zwherein f. generating a sequence of sets, S S S, of electrical signalsrepresentative of estimates 2,, 1,;"2, z,,,, 2,, of said first rootwherein the set 5 of signals is generated in response to S to S to a setof electrical signals representative of f (z),,--,, and to a set ofelectrical signals representative of f (z),,.;, and wherein and applyingthe aforementioned processing steps to said another group of signals togenerate a set of electrical signals, representative of a first root ofand hence representative of a second root of f (z).

13. A method according to claim 12 wherein the step of generating a setS of electrical signals comprises:

a. generating in response to said group of electrical signals,

a set S of electrical signals representative of the centroid, zc, of thevalue of the roots of f (z), b. generating in response to said group ofelectrical signals and to said set 8,, of signals, a set 5 of electricalsignals representative of the mean radius, Rm, of the roots of f (z)about and c generating in response to said set 5,, of signals and tosaid set S of signals the set of electrical signals S representative of2,. wherein I2 zcl Rm.

and

b. generating in response to said set S and said set S the set S ofelectrical signals.

The method according to claim 12 wherein said step of generating anelectrical condition comprises:

a. generating, in response to the set S and S of signals, a set of testelectrical signals representative of whether the variation from the setS of signals to the set S meets a predetermined criterion, and

. generating, in response to the sets S and S of signals and to said setof test electrical signals, said electrical condition representative ofwhether both said variation and the change in the set S, of signals fromthe set S21?1 meet said predetermined criterion and thereby whether saidchange is within said predetermined limits.

16. The method of generating physical root conditions for a function f(z) of a single complex variable quantity 1 which comprises:

a. applying signals representing said function to means for generating afirst sequence of physical conditions within a computer, each of saidconditions representing approximations of said root quantity accordingto the recursive relation where n is the index of the condition in saidsequence,

b. terminating said sequence in-a first root condition, and

c. applying modified function signals to means for successivelygenerating in said computer subsequent sequences of said conditionsafter modifying prior to each sequence the function f (z) by dividingthe condition representative thereof by a condition representative ofthe product of the differences between the condition upon which thefunction f (z) depends and the previously stored end root conditions,namely,

i= l,2,3,4,...n.

17. The method of generating electrical root signals for a function f(z) of a single complex variable quantity z which comprises:

a. generating a first sequence of electrical signals within a computereach representing approximations of said root quantity according to therecursive relation where n is the index of the electrical signals insaid sequence.

b. terminating said sequence in a first electrical root signal.

and

c. successively generating in said computer subsequent sequences of saidsignalsafter modifying prior to each sequence the function f (z) bydividing the signal representative thereof by a signal representative ofthe product of the differences between the signal upon which thefunction f (2) depends and the previously stored electrical end rootsignals, namely i=l,2,3,4,...n.

18. A system of generating physical conditions representative of complexroots of a function f (z) of a single complex variable quantity z whichcomprises:

a. a generator for producing a series of plural number sequences ofelectrical physical conditions and including structure for generatingeach member of the sequence as to represent an approximation of a firstof said roots according to the recursive relation where n is the indexof the condition in said sequence.

b. means connected to said generator for terminating each said sequencein a root condition in response to a variation between one saidcondition and a next preceding condition which is within a preset limit,and

c. means responsive to said electrical physical conditions and operativein each sequence subsequent to the first sequence in said series formodifying the function f (z) in accordance with the relation 19. Asignal treating system for generating physical conditions related toroots of a complex function of a single independent variable conditionwhich comprises:

a. an evaluating unit for producing a pair of output conditions onoutput channels representing the values of the complex function inresponse to a pair of given starting conditions applied to test inputchannels of the evaluating unit,

b. a loop with two input channels and two output channels coupled to theoutput channels and input channels, respectively, of the evaluatingunit,

0. means for producing in said output channels of said loop a pair ofdifferent sets of conditions which are representative of complex numbersfor application to the evaluating unit input channels,

d. control means including means for generating successive- 1y iterativesolution conditions of said complex function representative of complexnumbers where said loop has structure according to the relation wherez,, 1,, are successive sets of system output conditions, said controlmeans having a control unit means for comparing each system outputcondition with the next preceding output condition and for causing astop in operation of the system when the change between successivesystem output conditions is within predetermined limits and forrestarting the system to run to succeeding stops, and

e. means connected with said evaluating unit for modifying all outputconditions from said evaluating unit after the first said stop independence upon the magnitude of each of the system output conditionsnext preceding each said stop in accordance with the relation K singlecomplex variable condition depends and the previously stored rootconditions. i z 21. A system for generating voltages representative ofthe complex roots of a function, which function varies in depen- 5 denceupon a single complex variable. which comprises:

a. means for storing a pair of voltages which represent differentestimates ofa first root ofsaid function. b. means for generatingdifferent function voltages representative of the value of said functionfor each of the i= l.2,3,4. ...n.

20v A system for generating conditions representative of the complexroots of a function. which function varies in dependence upon a singlecomplex variable condition, which comprises:

a. means for storing a pair of conditions which represent difl Storedvoltages fel'em estimates Ofa fihrst Ofsaid function 0. means forgenerating new root voltages including means means l generatingdlfferfmt l Condmons for subtracting from the product of the second tobe obrepresentat ve of the value of said function for each of the minedf Said mot voltages and the fi to be obtained of Stored condmons- I saidfunction voltages, the product of the first used of said mcans forgenermng new root Condmons mcludmg root voltages and the second used ofsaid function voltmeans f subtraftmg from product of the Second to ages,and for dividing the difference by the difference of confjfuons and thefirst to be between the second of said function voltages subtractedtamed of said function conditions, the product of the first from thefirst of Said function voltages used P 99 commons i used of and d. afeedback system for alternately replacing the members functionconditions, and for dividing the difference by the ofsaid pair withsuccessive new root voltages 1 difference between the second ofsaidfunction conditions e means for comparing each new mot voltages with asubtracted from the first of said function conditions, preceding rootvoltage and operative when the $2 5??? a fg g ii gtg gj g 2513 95 1members ference between the penultimate and end stored root P voltagesis within a predetermined limit, to interrupt e. means for comparingeach new root condition with a generation Ofnew root voltages precedingroot condmon i operative when the f. means for separately storing theend root voltages, and ference between the penultimate and end storedroot g. means for initiating subsequent generation of new rootconditions is within a predetermined limit, to interrupt l I Igeneration of new mot conditions, vo tages ea mg to a itional end rootvo tages wherein f. means for separately storing each end rootcondition, and means are i g i for z each i g g. means for initiatingsubsequent generation of new root g i g i? 0 i lonlo conditions leadingto additional end root conditions and an 5. 8 age VI mg Sm e s a ex "3 1including means for modifying each said function condiy a 3 i fU o t e 9tion generated subsequent to the generation of the first e erencesefween t f upon w i end root condition by dividing said single complexvariasmgle complex z l depends and the Prev" ble condition by acondition representative of the product Ously Stored root con of thedifferences between the condition upon which said

1. The method of generating physical conditions representing complexroots of a function f (z) of a single complex variable quantity z whichcomprises: a. generating a first sequence of physical conditions withina computer each representing approximations of a first of said rootsaccording to the recursive relation where n is the index of theconditions in said sequence, b. terminating said sequence in a firstroot condition, and c. successively generating in said computersubsequent sequences of said conditions to produce the second, third,..., nth root conditions, where in the case of each sequence leading toa root condition, prior to each sequence the function f (z) is modifiedby dividing the condition representative thereof by a conditionrepresentative of the product of the differences between the conditionupon which the function f (z) depends and the previously stored end rootconditions, namely, i 1, 2, 3, 4, ... n.
 2. The method of generatingconditions representative of the complex roots of a function, whichfunction varies in dependence upon a single complex variable condition,which comprises: a. inputting to a computer a pair of physicallyrepresented complex root conditions which represent different estimatesof a first root of said function, b. generating in said computerdifferent physical conditions representative of the value of saidfunction for each of the stored conditions, c. alternately replacingsaid complex root condition with successive root conditions produced bysubtracting from the product of the second of said root conditions andthe first of said function conditions the product of the first of saidroot conditions and the second of said function conditions, and dividingthe difference by the difference between the second of said functionconditions subtracted from the first of said function conditions, untilthe difference between the penultimate and end stored root conditions isbelow a predetermined limit, d. separately storing the end rootcondition, and e. similarly generating in said computer additional endroot conditions with each said function condition generated subsequentto the generation of the first end root condition modified by dividingeach function condition by a condition representative of the product ofthe differences between the condition upon which the function conditiondepends and previously stored end root conditions.
 3. The method ofclaim 2 in which each starting pair of root conditions is representativeof points spaced from the centroid of all unknown root conditions adistance of the order of the mean radius of said unknown rootconditions.
 4. The method of claim 2 in which each starting pair of rootconditions is generated in dependence upon the coefficients of elementsof said function closely to approximate end root conditions as startingconditions.
 5. The method of claim 2 in which the root conditions ofsaid pair are generated to have values in the region of the centroid ofall of the roots of said function.
 6. The method of generating a pair ofvoltages representative of the complex roots of a function, whichfunction varies in dependence upon a single complex variable voltage,which comprises: a. inputting to a computer a pair of complex rootvoltages which represent different estimates of a first root of saidfunction, b. generating in said computer different function voltagesrepresentative of the value of said function for each of said rootvoltages, c. alternately replacing said root voltages with successiveroot voltages produced by subtracting from the product of the second ofsaid root voltages and the first of said function voltages, the productof the first of said root voltages and the second of said functionvoltages, and dividing the difference by the difference between thesecond of said function voltages subtracted from the first of saidfunction voltages, until the difference between the penultimate and endstored root voltages is sufficiently small, d. separately storing theend root voltage, and e. thereafter generating additional end rootvoltages with each said function voltage generated subsequent to thegeneration of the first end root voltage modified by dividing saidsingle complex variable voltage by a voltage representative of theproduct of the differences between the condition upon which said voltagedepends and the previously stored root voltages.
 7. The method ofgenerating conditions representative of the complex roots of a function,which function varies in dependence upon a single complex variablecondition, which comprises: a. inputting to a computer a pair of complexroot conditions which represent different estimates of a first root ofsaid function, b. generating in said computer different functionconditions representative of the value of said function for each of theroot conditions, c. alternately replacing the members of said pair withsuccessive root conditions produced by subtracting from the product ofthe second of said root conditions and the first of said functionconditions, the product of the first of said root conditions and thesecond of said function conditions, and dividing the difference by thedifference between the second of said function conditions subtractedfrom the first of said function conditions, until the difference betweenthe penultimate and end stored root conditions is within predeterminedlimits, d. separately storing the end root condition, and e. thereaftergenerating additional end root conditions with each said functioncondition generated subsequent to the generation of the first end rootcondition modified by dividing said single complex variable condition bya condition representative of the product of the differences between thecondition upon which said single complex variable condition depends andthe previously stored root conditions.
 8. The method of generatingconditions representative of the complex poles of a function, whichfunction varies in dependence upon a single complex variable condition,which comprises: a. inputting to a computer a pair of complex poleconditions which represent different estimates of a first pole of saidfunction, b. generating in said computer different function conditionsrepresentative of the value of said function for each of the rootconditions, c. alternately replacing the members of said pair withsuccessive pole conditions produced by subtracting from the product ofthe second of said pole conditions and the second of said functionconditions, the product of the first of said pole conditions and thefirst of said function conditions, and dividing the difference by thedifference between the first of said function conditions subtracted fromthe second of said function conditions, until the difference between thepenultimate and end stored pole conditions is sufficiently small, d.separately storing the end pole condition, and e. thereafter generatingadditional end pole conditions with each said function conditiongenerated subsequent to the generation of the first end pole conditionmodified by dividing said single complex variable condition by acondition representative of the product of the differences between thecondition upon which said single complex variable condition depends andthe previously stored root conditions.
 9. The method of generatingphysical conditions representing complex roots of a function f (z) of asingle complex variable quantity z which comprises: a. generating afirst sequence of physical conditions each representing approximationsof a first of said roots according to the recursive relation where n isthe index of conditions in said sequence, b. terminating said sequencein a first root condition, and c. generating subsequent sequences ofsaid conditions to produce the second, third, ..., nth root conditions,where in the case of each sequence leading to a root condition, and innumber corresponding with the order of said function f (z) withelimination in each said subsequent sequence of the effect of allearlier generated root conditions by dividing the conditionrepresentative of f (z) by a condition representative of the product ofthe difference between said condition upon which f (z) depends and thepreviously stored end root conditions.
 10. The method of generatingphysical conditions representing complex roots of a function f (z) of asingle complex variable quantity z which comprises: a. generating afirst sequence of physical conditions each representing approximationsof a first of said roots according to the recursive relation where n isthe index of conditions in said sequence, b. terminating said sequencein a first root condition, and c. successively generating subsequentsequences of said conditions to produce the second, third, ..., nth rootconditions, where in the case of each sequence leading to a rootcondition, with modification in each said subsequent sequence of theconditions representative of the quantities f (z)n-1 and f (z)n-2 bydividing the condition representative of the quantity f (z)n-1 by theproduct 1(z-1) and by dividing f (z)n-2 by a condition representative ofthe quantity 2(z-2) to eliminate the effect of all earlier generatedroot conditions.
 11. The method of producing physical representations ofthe roots of a complex function f (z) which comprises: a. generating afirst set of starting conditions z1 representative of a first estimateof the location of a first root, b. generating a second set of startingconditions z2 representative of a second estimate of the location ofsaid first root, c. generating a third set of starting conditions z3representative of a third and improved description of the location ofsaid first root according to the relation where n 3, d. thereaftersubstituting members of the series z3, z4, ..., zn for representationsof z1 and z2 in the sequence z3 for z1, z4 for z2, z5 for z3 ..., untilthe change between conditions representing zn-1 and zn is within thepredetermined limits whereby conditions zn represent said first root,and e. repeating the aforementioned steps with successive modificationof the conditions representative of the functions f (z)n-2, f (z)n-3..., the modification being in accordance with the relation i 1, 2, 3,4, ... n to eliminate the effect Therein of root functions previouslygenerated.
 12. The method of processing a group of electrical signalsrepresentative of values of a complex function f (z) to produceelectrical signals representative of the value of roots of said functionwhich comprises: a. generating a set Sz of electrical signals,representative of a first estimate z1 for the first of said roots, b.generating a set Sz of electrical signals, representative of a secondestimate z2 for said first of said roots, c. generating, in response tosaid group of signals and to said set Sz of signals, a set Sf(z) ofelectrical signals, representative of a value f (z)1, d. generating, inresponse to said group of signals and to said set Sz of signals, a setSf(z) of electrical signals, representative of f(z)2, e. generating, inresponse to said set Sz , Sz , Sf(z) , Sf(z) , of electrical signals, aset Sz of electrical signals, representative of an estimate z3, whereinf. generating a sequence of sets, Sz , ..., Sz , Sz of electricalsignals representative of estimates z1, ..., zn-2, zn-1, zn of saidfirst root wherein the set Sz of signals is generated in response to Sz, to Sz , to a set of electrical signals representative of f (z)n-1, andto a set of electrical signals representative of f (z)n-2, and whereing. generating in response to the sets Sz and Sz of signals, anelectrical condition representative of whether the change in the set Szof signals from the set Sz of signals is within predetermined limits, h.terminating the generation of said sequence, in response to saidelectrical condition, when said condition is indicative that said changeis within said limits, and i. generating, in response to said group ofelectrical signals and to said set Sz , another group of electricalsignals representative of and applying the aforementioned processingsteps to said another group of signals to generate a set of electricalsignals, representative of a first root of and hence representative of asecond root of f (z).
 13. A method according to claim 12 wherein thestep of generating a set S1 of electrical signals comprises: a.generating in response to said group of electrical signals, a set Szc ofelectrical signals representative of the centroid, zc, of the value ofthe roots of f (z), b. generating in response to said group ofelectrical signals and to said set Szc of signals, a set SRm ofelectrical signals representative of the mean radius, Rm, of the rootsof f (z) about zc and c. generating in response to said set Szc ofsignals and to said set SRm of signals the set of electrical signals Szlrepresentative of z1, wherein
 14. The method according to claim 12wherein the step of generating the set Sz of electrical signals includesthe steps of: a. generating in response to said sets Sz , Sz , Sf(z) ,and Sf(z) , the set S(z -z ) of electrical signals representative of thevalue of the quantity and b. generating in response to said set S(z -z )and said set Sz , the set Sz of electrical signals.
 15. The methodaccording to claim 12 wherein said step of generating an electricalcondition comprises: a. generating, in responSe to the set Sz and Sz ofsignals, a set of test electrical signals representative of whether thevariation from the set Sz of signals to the set Sz meets a predeterminedcriterion, and b. generating, in response to the sets Sz and Sz ofsignals and to said set of test electrical signals, said electricalcondition representative of whether both said variation and the changein the set Sz of signals from the set Sz meet said predeterminedcriterion and thereby whether said change is within said predeterminedlimits.
 16. The method of generating physical root conditions for afunction f (z) of a single complex variable quantity z which comprises:a. applying signals representing said function to means for generating afirst sequence of physical conditions within a computer, each of saidconditions representing approximations of said root quantity accordingto the recursive relation where n is the index of the condition in saidsequence, b. terminating said sequence in a first root condition, and c.applying modified function signals to means for successively generatingin said computer subsequent sequences of said conditions after modifyingprior to each sequence the function f (z) by dividing the conditionrepresentative thereof by a condition representative of the product ofthe differences between the condition upon which the function f (z)depends and the previously stored end root conditions, namely,
 17. Themethod of generating electrical root signals for a function f (z) of asingle complex variable quantity z which comprises: a. generating afirst sequence of electrical signals within a computer each representingapproximations of said root quantity according to the recursive relationwhere n is the index of the electrical signals in said sequence, b.terminating said sequence in a first electrical root signal, and c.successively generating in said computer subsequent sequences of saidsignals after modifying prior to each sequence the function f (z) bydividing the signal representative thereof by a signal representative ofthe product of the differences between the signal upon which thefunction f (z) depends and the previously stored electrical end rootsignals, namely i 1, 2, 3, 4, ... n.
 18. A system of generating physicalconditions representative of complex roots of a function f (z) of asingle complex variable quantity z which comprises: a. a generator forproducing a series of plural number sequences of electrical physicalconditions and including structure for generating each member of thesequence as to represent an approximation of a first of said rootsaccording to the recursive relation where n is the index of thecondition in said sequence, b. means connected to said generator forterminating each said sequence in a root condition in response to avariation between one said condition and a next preceding conditionwhich is within a preset limit, and c. means responsive to saidelectrical physical conditions and operative in each sequence subsequentto the first sequence in said series for modifying the function f (z) inaccordance with the relation i 1, 2, 3, 4, ... n.
 19. A signal treatingsystem for generating physical conditions related to roots of a complexfunction of a single independent variable condition which comprises: a.an evaluating unit for producing a pair of output conditions on outputchannels representing the values of the complex function in response toa pair of given starting conditions applied to test input channels ofthe evaluating unit, b. a looP with two input channels and two outputchannels coupled to the output channels and input channels,respectively, of the evaluating unit, c. means for producing in saidoutput channels of said loop a pair of different sets of conditionswhich are representative of complex numbers for application to theevaluating unit input channels, d. control means including means forgenerating successively iterative solution conditions of said complexfunction representative of complex numbers where said loop has structureaccording to the relation where z1, z2, ..., zn are successive sets ofsystem output conditions, said control means having a control unit meansfor comparing each system output condition with the next precedingoutput condition and for causing a stop in operation of the system whenthe change between successive system output conditions is withinpredetermined limits and for restarting the system to run to succeedingstops, and e. means connected with said evaluating unit for modifyingall output conditions from said evaluating unit after the first saidstop in dependence upon the magnitude of each of the system outputconditions next preceding each said stop in accordance with the relationi 1, 2, 3, 4, ... n.
 20. A system for generating conditionsrepresentative of the complex roots of a function, which function variesin dependence upon a single complex variable condition, which comprises:a. means for storing a pair of conditions which represent differentestimates of a first root of said function, b. means for generatingdifferent function conditions representative of the value of saidfunction for each of the stored conditions, c. means for generating newroot conditions including means for subtracting from the product of thesecond to be obtained of said root conditions and the first to beobtained of said function conditions, the product of the first used ofsaid root conditions and the second used of said function conditions,and for dividing the difference by the difference between the second ofsaid function conditions subtracted from the first of said functionconditions, d. a feedback system for alternately replacing the membersof said pair with successive new root conditions, e. means for comparingeach new root condition with a preceding root condition and operative,when the difference between the penultimate and end stored rootconditions is within a predetermined limit, to interrupt generation ofnew root conditions, f. means for separately storing each end rootcondition, and g. means for initiating subsequent generation of new rootconditions leading to additional end root conditions and including meansfor modifying each said function condition generated subsequent to thegeneration of the first end root condition by dividing said singlecomplex variable condition by a condition representative of the productof the differences between the condition upon which said single complexvariable condition depends and the previously stored root conditions.21. A system for generating voltages representative of the complex rootsof a function, which function varies in dependence upon a single complexvariable, which comprises: a. means for storing a pair of voltages whichrepresent different estimates of a first root of said function, b. meansfor generating different function voltages representative of the valueof said function for each of the stored voltages, c. means forgenerating new root voltages including means for subtracting from theproduct of the second to be obtained of said root voltages and the firstto be obtained of said function voltages, the product of the first usedof said root voltages and the second used of said function voltages, andfor dividing the difference by the difference between the second of saidfunction voltages subtracted from the first of said function voltages,D. a feedback system for alternately replacing the members of said pairwith successive new root voltages, e. means for comparing each new rootvoltages with a preceding root voltage and operative, when thedifference between the penultimate and end stored root voltages iswithin a predetermined limit, to interrupt generation of new rootvoltages, f. means for separately storing the end root voltages, and g.means for initiating subsequent generation of new root voltages leadingto additional end root voltages wherein means are provided for modifyingeach said function voltage generated subsequent to the generation of thefirst end root voltage by dividing said single complex variablecondition by a condition representative of the product of thedifferences between the condition upon which said single complexvariable condition depends and the previously stored root conditions.